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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 2nd 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexon my personal web I am starting a page [[schreiber:derived critical locus]] with some notes. I think so far I can convince myself of the claim that the page currently ends with (without proof). My next goal is to show that the homotopy fibers discussed there are given by BV-BRST complexes. But I have to interrupt now.

   on my personal web I am starting a page derived critical locus (schreiber) with some notes.

   I think so far I can convince myself of the claim that the page currently ends with (without proof). My next goal is to show that the homotopy fibers discussed there are given by BV-BRST complexes. But I have to interrupt now.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 9th 2011
   • (edited Mar 9th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexmaybe I made some progress with understanding the BV-complexes formally as derived critical loci: as homotopy fibers of sections $d S : \mathfrak{a} \to T^* \mathfrak{a}$ of the cotangent bundle on an $\infty$-Lie algebroid $\mathfrak{a}$ (a formal dual to a BRST complex). New, rewritten notes are at [[schreiber:derived critical locus]]. In fact I think I understand the full story if I assume that homotopy pushouts of my unbounded commutative dg-algebras are computed by mapping cones as usual. This is what one expects, but one needs to be a bit careful with what model structure exactly one uses to present the derived geometry, and what assumptions on projectivity are being made. This I need to think more about.

   maybe I made some progress with understanding the BV-complexes formally as derived critical loci: as homotopy fibers of sections $d S : \mathfrak{a} \to T^* \mathfrak{a}$ of the cotangent bundle on an $\infty$-Lie algebroid $\mathfrak{a}$ (a formal dual to a BRST complex).

   New, rewritten notes are at derived critical locus (schreiber).

   In fact I think I understand the full story if I assume that homotopy pushouts of my unbounded commutative dg-algebras are computed by mapping cones as usual. This is what one expects, but one needs to be a bit careful with what model structure exactly one uses to present the derived geometry, and what assumptions on projectivity are being made. This I need to think more about.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeSep 27th 2011
   • (edited Sep 27th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexNew entry [[derived critical locus]] at the main $n$Lab to record the Vezzosi's paper. I am a bit surprised that the page lists that it is linked from [[derived critical locus]] while I have not put that self-referencing link.

   New entry derived critical locus at the main $n$Lab to record the Vezzosi’s paper. I am a bit surprised that the page lists that it is linked from derived critical locus while I have not put that self-referencing link.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 28th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexOh, wow. Thanks, I had not seen that.

   Oh, wow.

   Thanks, I had not seen that.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 1st 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have brought into _[[derived critical locus]]_ the core of my [old notes](https://ncatlab.org/schreiber/show/derived+critical+locus) (from my personal web) aiming to show that the BV-BRST complex really is (the formal dual of) the derived critical locus in [[dg-geometry]] of a function on a Lie algebroid (BRST complex). Looking at this material from 2011 again I notice two things: 1. In the example I don't check the smoothness assumption made in [this prop.](https://ncatlab.org/nlab/show/derived+critical+locus#PresentationByFreeDGCAlegbraOnMappingCone) 1. meanwhile there ought to be a reference that provides all the required model-category theoretic background in the entry in an easily citable way. I don't quite have the time right now to dig into this again. If anyone has a hint, I'd be grateful.

   I have brought into derived critical locus the core of my old notes (from my personal web) aiming to show that the BV-BRST complex really is (the formal dual of) the derived critical locus in dg-geometry of a function on a Lie algebroid (BRST complex).

   Looking at this material from 2011 again I notice two things:

   1. In the example I don’t check the smoothness assumption made in this prop.

   2. meanwhile there ought to be a reference that provides all the required model-category theoretic background in the entry in an easily citable way.

   I don’t quite have the time right now to dig into this again. If anyone has a hint, I’d be grateful.

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 2nd 2017
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexThe Costello-Gwilliam reference seems to now be [this book](https://pdfs.semanticscholar.org/0eb5/652340bb414869764cf5b30feed90597558d.pdf) (pdf); does anyone have a more precise location for the claim referred to in the nLab page? Or is it just the general philosophy of the approach?

   The Costello-Gwilliam reference seems to now be this book (pdf); does anyone have a more precise location for the claim referred to in the nLab page? Or is it just the general philosophy of the approach?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeOct 2nd 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItex> a more precise location for the claim In the book it is now the beginning of section 4.8.1 (Back in 2011 I was pointing to their wiki, which however no longer exists. I have added the section pointer to the entry now.)

   a more precise location for the claim

   In the book it is now the beginning of section 4.8.1

   (Back in 2011 I was pointing to their wiki, which however no longer exists. I have added the section pointer to the entry now.)

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeOct 2nd 2017
   • (edited Oct 2nd 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexVincent Schlegel kindly pointed out to me that, as stated, the computation gave the critical locus in $C \times \mathbb{A}^1$ instead of in $C$, while the latter is really the further pullback along $C \to C \times \mathbb{A}^1$. I have fixed this. Dually the point is that in $Sym_{\mathcal{O}(C)}\left( \mathcal{O}(C) \oplus \cdots \right) $ there are "two copies" of $\mathcal{O}(C)$, and they eventually need to be identified. Indeed that's what is necessary to yield the desired conclusion (which tacitly made that identification). I have fixed this now.

   Vincent Schlegel kindly pointed out to me that, as stated, the computation gave the critical locus in $C \times \mathbb{A}^1$ instead of in $C$, while the latter is really the further pullback along $C \to C \times \mathbb{A}^1$. I have fixed this.

   Dually the point is that in $Sym_{\mathcal{O}(C)}\left( \mathcal{O}(C) \oplus \cdots \right)$ there are “two copies” of $\mathcal{O}(C)$, and they eventually need to be identified. Indeed that’s what is necessary to yield the desired conclusion (which tacitly made that identification).

   I have fixed this now.

9. • CommentRowNumber9.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 2nd 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexRe #6, we have a dedicated page [[Factorization algebras in perturbative quantum field theory]], which points to two volumes.

   Re #6, we have a dedicated page Factorization algebras in perturbative quantum field theory, which points to two volumes.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2017
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks. I have made the reference line point to that page. I'd have to check which volume is relevant.

    Thanks. I have made the reference line point to that page. I’d have to check which volume is relevant.

11. • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 3rd 2017
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexThanks, Urs.

    Thanks, Urs.